A polyhedron operation that preserves unicursality for general face spectra

In polyHédronisme, this is t3 n(10,0.5,0.3) n(5,0.5,0.3) n(3,0.5,0.3) n(4,0.5,0.3) J20. It is unicursal just like its base polyhedron, Johnson solid J20.

The previously described polyhedron operations are limited to certain constraints on vertex valence or face valence. The operation described here can deal with any face size. The idea is to augment every face with its frustum and then truncate the new vertices. This can be accomplished with polyHédronisme's set of operators though the expression must be tailored to the range of face valencies in the base polyhedron. Since the frustum augmentation step (it is called inset, n, in polyHédronisme) creates new 4-valent faces, the frustum operation on 4-valent faces must be done first. The frustum operations on {3, 5, 6, 7...etc.}-valent faces can then be done in any order. After all the faces have been inset, all of the original vertices are now even-valent, so none are valence 3, and polyHédronisme operation t3 only truncates the new vertices.

If 'n...n' can stand for any number of valence-specific n's, this operation is t3n...n4.

Demonstration that t3n...n4 is unicursal for any face size: l to r, triple every edge in the graph (inverse III); doubly subdivide the arcs internal to each face (inverse IV); connect these new vertices peripherally with digons (inverse III); delta-to-Y the triangles thus created (V).